Chi distribution

chi

Probability density function
Cumulative distribution function
Parameters	${\displaystyle k>0\,}$ (degrees of freedom)
Support	${\displaystyle x\in [0;\infty )}$
PDF	${\displaystyle {\frac {2^{1-k/2}x^{k-1}e^{-x^{2}/2}}{\Gamma (k/2)}}}$
CDF	${\displaystyle P(k/2,x^{2}/2)\,}$
Mean	${\displaystyle \mu ={\sqrt {2}}\,{\frac {\Gamma ((k+1)/2)}{\Gamma (k/2)}}}$
Mode	${\displaystyle {\sqrt {k-1}}\,}$ for ${\displaystyle k\geq 1}$
Variance	${\displaystyle \sigma ^{2}=k-\mu ^{2}\,}$
Skewness	${\displaystyle \gamma _{1}={\frac {\mu }{\sigma ^{3}}}\,(1-2\sigma ^{2})}$
Ex. kurtosis	${\displaystyle {\frac {2}{\sigma ^{2}}}(1-\mu \sigma \gamma _{1}-\sigma ^{2})}$
Entropy	${\displaystyle \ln(\Gamma (k/2))+\,}$ ${\displaystyle {\frac {1}{2}}(k\!-\!\ln(2)\!-\!(k\!-\!1)\psi _{0}(k/2))}$
MGF	Complicated (see text)
CF	Complicated (see text)

In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the square root of the sum of squares of independent random variables having a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables from the origin. The most familiar examples are the Rayleigh distribution with chi distribution with 2 degrees of freedom, and the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom (one for each spatial coordinate). If ${\displaystyle X_{i}}$ are k independent, normally distributed random variables with means ${\displaystyle \mu _{i}}$ and standard deviations ${\displaystyle \sigma _{i}}$, then the statistic